Axiom Of Choice

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 102 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
123 \(\Rightarrow\) 62	Two model theoretic ideas in independence proofs, Pincus, D. 1976, Fund. Math.
62 \(\Rightarrow\) 102	The Axiom of Choice, Jech, 1973b, page 162 problem 11.12

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
123:	\(SPI^\): Uniform weak ultrafilter principle:* For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
102:	For all Dedekind finite cardinals \(p\) and \(q\), if \(p^{2} = q^{2}\) then \(p = q\). Jech [1973b], p 162 prob 11.12.

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